For $p$ & $q$, why do we need the condition- $a_{1}b_{2}−a_{2}b_{1}$ to be invertible, to assert that $gcd(p,q)=gcd(a_{1}p+b_{1}q,a_{2}p+b_{2}q)$?

Question

For $p$ & $q$, why do we need the condition- $a_{1}b_{2}−a_{2}b_{1}$ to be invertible, to assert that $gcd(p,q)=gcd(a_{1}p+b_{1}q,a_{2}p+b_{2}q)$?

Here, $p$ & $q$ are polynomials and $a_{1}, b_{1}, a_{2}, b_{2}$ are any scalars.

Also, what do we mean by an expression to be invertible here?

NB- The statement in the title of the question was encountered when I was reading Wikipedia.